 
Summary: University of Washington Math 523A Lecture 2
Martingales: concentration I
Lecturer: Eyal Lubetzky
April 3, 2009
The HoeffdingAzuma concentration inequality
Theorem 1 (Hoeffding `63). Let (Xt) be a martingale with respect to (Ft), and let c1, c2, . . . be
real numbers such that Xt  Xt1 ct for all t. Then for any a > 0,
P(Xn  X0 a) 2 exp 
a2
2 n
i=1 c2
i
.
Proof. We will show a slightly stronger version of the above theorem: Instead of requiring that
(Xt) is a martingale, we will assume that
E[Xt  Xt1] = Xt1 for all t . (0.1)
Let Yt = Xt  Xt1. By the above assumptions E[Yt  Xt1] = 0 and Yt ct for all t.
The following simple claim says that, if Z is a r.v. bounded by 1 and with mean 0, then the
expected value of exp(Z) is maximized when splitting the mass equally between ±1.
Claim 2. Let Z be a random variable satisfying Z 1 and EZ = 0. Then
